Dirac tensor with heavy photon
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UCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Dirac Tensor with Heavy Photon1 V. V. Byteva, E. A. Kuraeva, and E. S. Scherbakovab a
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia b Hamburg University, 22767 Hamburg, Germany email: [email protected], [email protected] Received January 26, 2012
Abstract—For the largeangle hardphoton emission by initial leptons in the process of highenergy annihi lation of e+e– to hadrons, the Dirac tensor is obtained by taking the lowestorder radiative corrections into account. The case of largeangle emission of two hard photons by initial leptons is considered. In the final result, the kinematic case of collinear emission of hard photons and soft virtual and real photons is included; it can be used for the construction of MonteCarlo generators. DOI: 10.1134/S1063776113030047 1
1. INTRODUCTION In the experiments with e+e– annihilation to hadrons, the important role is played by the so called “returning to resonance” mechanism. It con sists in the emission of a hard real photon by initial leptons [1]. The Born contribution and the oneloop correc tion are taken into account in the Dirac tensor (the crosssymmetry partner of the Compton tensor—a bilinear combination of the hard photon emission cur rents averaged over lepton spin states and summed over photon polarization states). Infrared divergences are parameterized by introducing the “photon mass λ. In the final expression, it is removed in a usual way by adding the contribution from additional soft photon emission. We do not consider photon emission by the final charged particles and the effects of chargeodd inter ference of virtual or real photon emission from leptons and hadrons. Therefore, the Dirac tensor obtained in this way is universal. The paper is organized as follow. In Section 2, the relation of the Dirac tensor to the cross section of radi ative annihilation of a lepton pair to hadrons is clari fied. We give the Bornlevel expression for the Dirac tensor and derive the general form of the radiative cor rection to it using the symmetry relation. In Section 3, we obtain the contribution arising from the mass oper ator of the positron and vertex function in the case where a positron and a photon are on the mass shell. In Section 4, we consider the contribution from the ver tex function to the case of an oilshell electron and the boxtype Feynman amplitude with an electron, positron, and one of the photons on the mass shell. In Section 5, we analyze the total result for the Dirac ten sor, adding the emission of additional soft photon con 1
tributions, which provide the final result that is free from the infrared divergences. The limit case of an almost collinear hard photon emission is considered and some numerical estimates are given. We give the hadronic tensor for several final states: γ* π+π–, μ+μ–, ρ+ρ–. In Appendices A and B, the details of the calculation are presented. In Appendix C, the contribution to
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